Bimodal condensates (I)

The third lesson dealt with the description of bimodal condensates, degenerate gas systems in which the atoms are distributed in two quantum levels. We began by recalling how to describe a condensate in second quantization. We then introduced the coherent states of a bimodal condensate in the absence of interaction. For a coherent bimodal condensate, we described the fluctuation in the partition of the number of bosons in each of the two levels. We then discussed the effect of interactions. It leads to a freezing of fluctuations when the interaction is repulsive, and to the appearance of a Schrödinger cat-like ground state when it is attractive. We have shown how these states could, in principle, be prepared. Finally, we compared the properties of coherent states, frozen-fluctuation states and cat-like states. The second part of the lesson was devoted to the oscillations of a bimodal condensate in a double potential well, which can be thought of as the "Josephson effect of cold atoms". We gave the expression for the probability current across the barrier separating the two wells. We have calculated the oscillation frequency using two models, one microscopic, the other macroscopic. We have heuristically established the expression of the Josephson Hamiltonian associating as conjugate variables the phase difference and the population difference between the wells. We gave the solution describing the dynamic evolution governed by this Hamiltonian, in the case of small oscillations by defining the plasma frequency. We have shown that populations tend to self-trapping in the case of large initial imbalances. We described an experiment that allowed us to observe these oscillations and their blocking.